The flange bent hardest where it curls least

The flange that curls away from its stress followed a curved I-beam’s flange as it carried its stress round a bend. A flange stressed along a curve is pressed sideways by that stress, towards the centre of its own curvature. Each outstand bends like a cantilever from the web, and in bending it moves to a different radius and sheds stress. The whole of it was governed by one number, $\lambda = [3(1-\nu^2)]^{1/4}\,b/\sqrt{Rt}$, with $b$ the outstand, $R$ the radius and $t$ the thickness. The effective width fell slowly as λ grew. The stress across the flange at its root — a cantilever’s root bending stress, acting across the beam rather than along it — arrived at once, equal to the longitudinal stress by λ = 0.8 and tending to 1.82 times it.

Every flange there kept its outstand and its thickness all round the curve, so one λ described it — as the wide side goes inside found for a hook’s section, the shape was doing two things at once and one λ was enough to say both. That essay ended on the commonest tightly curved I-section in building, the knee of a portal frame, whose flanges are cut from plate to a curve and usually taper — in width, in thickness, or both — from the rafter to the column, the corner where the frame that leans gets its stiffness against sway and its largest moment. In a tapered knee λ changes along the curve. The question left was whether the transverse stress then peaks where λ is largest, or where the effective width changes fastest and force is moved between the flange and the web.

A knee, and four flanges #

The knee is curved to 1,000 mm at its inner flange through a quarter turn, a curve 1.57 m long, with the flanges 600 mm apart on a 10 mm web. It carries 500 kN·m all round the curve, which isolates the taper: whatever changes along the knee changes because the plate does, not because the moment does. The outer flange, at a radius of 1,600 mm, has the same plate as the inner one. It curls too, but less: for the uniform plate its λ is 0.90 against the inner flange’s 1.14, because the radius under the square root is larger. The inner flange is the one followed here because it is the tighter curve, and because at a portal knee under gravity it is the compression flange, on the inside of the corner, where a flange bent across its width at the root is also the flange that must not buckle.

Four inner flanges are compared, each running from the rafter end of the knee to the column end: one uniform at 250 mm wide and 20 mm thick; one narrowing from 280 to 180 mm at 20 mm thick; one thinning from 25 to 15 mm at 250 mm wide; and one doing both, from 280 × 25 to 180 × 15. The uniform flange and the tapered ones carry roughly the same steel, and the taper both ways is what a fabricator does to a knee whose flange must meet a narrower, thinner column flange at one end.

Every strip across the flange is solved as Bleich’s strip at its own local width and thickness. That treats the strips as independent of their neighbours along the curve, which is right when the taper is slow beside the strip’s own bending length, $\sqrt{Rt}$ — about 140 mm here, against a curve ten times as long. Each section of the knee is then a straight-beam section with its flanges at their effective widths, which gives the flange’s own stress at that section and the force it carries.

Narrowing uncurls the flange, thinning curls it more #

λ grows with the outstand and falls as the square root of the thickness, so the two tapers pull it opposite ways. A flange that narrows from 280 to 180 mm has shorter outstands at the column end and curls less there: λ falls from 1.27 to 0.82. A flange that thins from 25 to 15 mm at a constant width is more flexible at the column end and curls more: λ rises from 1.02 to 1.31. A flange that does both comes out between, with λ falling from 1.14 to 0.94, because the narrowing is the stronger of the two — λ goes with the outstand directly and with the thickness only as its square root.

The effective width follows λ. The uniform flange works over 78% of its width all round the curve. The narrowing flange works over 71% at its wide end and 92% at its narrow end; the thinning flange over 84% at its thick end and 70% at its thin end; the flange tapered both ways over 78% and 87%. On this measure alone the flange tapered both ways looks better at its column end than its rafter end — it is using more of what it has.

The same force on less steel #

What the width figure does not show is how much steel there is to use. The knee carries the same moment at every section, the lever arm between its flanges is the same, and so the inner flange carries nearly the same force all round the curve: 696 kN at the rafter end of the flange tapered both ways and 579 at the column end, the web taking up a little more of the moment as the flange gets smaller. That force is carried at the column end on a plate 180 × 15 working over 87% of its width — 2,350 mm² against 5,460 at the rafter end — so the flange’s stress rises from 127 N/mm² to 246.

The other tapers do the same in proportion to the steel they lose. The uniform flange’s stress is 167 N/mm² all round; the narrowing flange’s rises from 164 to 192, the thinning flange’s from 132 to 226. A taper changes how hard a flange curls, and it changes how hard the flange is stressed in the first place, and the second change is the larger.

The root follows the flange’s stress #

The transverse stress at the root is the flange’s own stress times the strip’s curling factor, the ratio of transverse to longitudinal stress that λ sets: 1.61 for the uniform flange, whose root is bent across at 269 N/mm² all round. For either taper alone the two factors move together. The narrowing flange curls least at its narrow end, and its stress rises only modestly there, so its root is bent hardest at the wide end, 283 N/mm², where λ is largest. The thinning flange curls most at its thin end and is most stressed there too, and its root reaches 397 N/mm² — past the yield stress of an S355 plate, from a stress that no straight-beam check of the knee would compute.

Tapered both ways, the two factors part. λ is largest at the rafter end and the curling factor with it, 1.61 against 1.32 at the column end. But the flange’s stress has nearly doubled from the rafter end to the column end, and the product rises: 205 N/mm² at the rafter end, 325 at the column end. The flange is bent hardest across its width at the end where it curls least. A designer who looked for the transverse stress where λ is largest would check the rafter end and find it 37% below the one that governs.

Why the two tapers part #

The result can be read off two proportionalities, without the strips. The flange’s force $F$ is nearly the same all round the knee, so its stress is the force over its working area, $\sigma_0 \approx F/(\rho\,b\,t)$, with $\rho$ the share of the width that works. The root’s transverse stress is that stress times the curling factor $\phi$, and both $\rho$ and $\phi$ depend on λ alone, which goes as $b/\sqrt{t}$ at a given radius. So

$$\sigma_{\text{across}} \approx \frac{F}{b\,t}\cdot\frac{\phi(\lambda)}{\rho(\lambda)}.$$

Narrowing the flange raises $F/bt$ in proportion to $1/b$ but lowers λ in proportion to $b$, and around λ = 1 the factor $\phi/\rho$ falls faster than $1/b$ rises: the narrow end is bent less. Thinning the flange raises $F/bt$ in proportion to $1/t$ and raises λ as well, so both factors grow together: the thin end is bent much more. Doing both multiplies a falling λ by a stress that rises as $1/bt$, and the stress wins, because a plate that loses a third of its width and two fifths of its thickness has lost three fifths of its area while its λ has fallen by only a sixth.

That is the same competition the section that changes along the span found in a tapered cantilever, whose worst section lies where the depth has doubled rather than where the moment is largest, because the capacity is moving too. Here what moves is the plate’s area, and what it competes with is its curling.

The root in both directions at once #

The transverse stress at the root does not act alone. The same point carries the flange’s longitudinal stress, and the steel there is stressed in two directions at once. Combined by the von Mises rule, the uniform flange’s root reaches 235 N/mm² all round the knee. The narrowing flange’s worst is 246 at its wide end; the flange tapered both ways, 294 at its thin end; the thinning flange, 345 at its thin end, within 3% of the yield stress of S355, where the flange’s longitudinal stress is only 226 N/mm². The check a designer makes on the flange’s longitudinal stress passes everywhere. The check nobody makes on its root is the one that governs, and it governs at the end of the knee where the plate was chosen to be smallest.

The force handed to the web is small #

The essay’s other candidate was the place where the effective width changes fastest, on the reasoning that a changing effective width moves force between the flange and the web, and the junction has to carry the move. It does, and the force is small. The flange’s force changes because its area does, but the moment and the lever arm do not, and the change is about a sixth of the force spread over a metre and a half of curve. In a 10 mm web beside the weld that is at most 12 N/mm² of shear, against a shear yield of 205.

That is the difference from the straight beams of the flange that is not all there, where shear lag makes a wide flange’s effective width a property of the span, and the shear flow into the flange from the web is the whole of its load. A curling flange’s effective width is a property of the curve and the plate, and its force is set by the moment, so a slowly changing plate moves the force between flange and web only slowly. A sudden change of plate — a flange spliced from a wide plate to a narrow one at the middle of the knee — would concentrate the same move into one bending length, and that is a detail to be checked at the splice, not a stress spread along the curve.

A tighter knee, a harder root #

The radius sets the scale of all of it. At a radius of 5,000 mm both flanges curl little and their roots carry 65 and 73 N/mm²; at 1,000, 269 and 325; at 500, 376 and 500, both past yield. The tapered flange is the worse of the two at every radius, and its worst section is always its thin end. Tightening the knee raises λ everywhere, and at the thin end, where the flange’s stress is highest, it raises a stress that was already the largest.

This is the arithmetic of a portal knee drawn tight to save headroom — the same corner whose haunched rafter, in the section that governs is inside the haunch, is checked at a station no formula names — and it has a cheap answer that does not depend on the radius: keep the flange’s thickness, and take the taper in the width. Where the column’s flange really is thinner than the rafter’s, the change of thickness belongs at the column end of the curve, past the knee, where the flange is straight and has nothing to curl about. A flange narrowed rather than thinned is less stressed at its narrow end, curls less there, and keeps its worst root stress at the wide end, where there is the most plate to carry it.

The column end, by hand #

At the column end the flange tapered both ways is 180 × 15, with outstands of 90 mm, half the plate, measured as the strip model measures them from the web’s centre line. Its curling parameter is $(3 \times 0.91)^{1/4} \times 90/\sqrt{1{,}000 \times 15} = 1.286 \times 0.735 = 0.94$, and Bleich’s strip at that λ gives an effective width of 87% and a curling factor of 1.32.

The flange carries about 579 kN on $0.87 \times 180 \times 15 = 2{,}350\ \mathrm{mm^2}$, a stress of 246 N/mm², and its root is bent across at $1.32 \times 246 = 325$ N/mm². At the rafter end the same flange, 280 × 25 with λ = 1.14, carries 696 kN on $0.78 \times 280 \times 25 = 5{,}460\ \mathrm{mm^2}$, 127 N/mm², and its root sees $1.61 \times 127 = 205$.

What the strips leave out #

The strips are independent along the curve. Each strip is solved as if its neighbours had the same plate. That is right for a taper slow beside the strip’s bending length and wrong at a step; at a splice the flange’s stiffness changes inside one bending length, and the plate bends there in two directions at once.

The section is a straight beam’s. The flange’s stress here comes from straight-beam bending with effective flanges. The bar that was bent before it was loaded showed that a curved bar’s stress is a hyperbola, larger at the inner fibre than straight-beam theory says, and for a knee whose radius is less than twice its depth that adds to the inner flange’s stress. It adds in the same proportion at both ends of the knee, so it does not move where the root is bent hardest.

The moment is the same all round. A real knee carries a moment that varies a little across it and is largest towards the column, which moves the worst section further towards the column end of a flange tapered that way.

The steel stays elastic. At the thin end of the thinning flange the root’s combined stress is within a few per cent of yield. A root that yields across the flange forms a line of plastic hinges along the toe of the flange-to-web weld; the outstands then turn more freely, curl further, and shed width faster than the elastic strips say, so the first yielding is also where the effective width begins to fall away. That is the failure the thin end of a tapered knee approaches first, and it approaches it from a stress the knee’s ordinary design check does not contain.

The web stays straight. The flange’s curling pulls on the web along the root, and a thin web bends in turn; the web that is crushed from inside followed a curved compression flange pressing into its web. A stiffener at the knee’s mid-point would hold the root and change the strip’s boundary there.

Still open: the knee with a stiffener across its flange #

A curved knee is often given a radial stiffener at its middle, welded across the inner flange and the web, to hold the flange against exactly this curling. A stiffener holds the outstands’ tips where it crosses them, so the strip there is no longer a cantilever from the web but a plate held on three sides, and the curling between stiffeners becomes a two-dimensional plate problem with the stiffener spacing as a second length. Whether one stiffener at the middle of a quarter-turn knee is enough to bring the thin end’s root stress under yield, or whether the stiffeners must be closer than a bending length to make any difference at all, is the question the strip model cannot answer and the moment that goes round the corner leaves at every knee that is curved instead of square.